What is modern algebraic topology(homotopy theory) about?

Question

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other areas of mathematics, especially, in geometry(I would say almost in all geometry).

I'm not an algebraic topologist myself, so I know only basic techniques. However, I'm intrigued by modern tool in homotopy theory. For example, we have simplicial homotopy theory, where one studies simplicial sets instead of topological spaces.

As far as I understand, simplicial techniques are indispensible in modern topology. Then we have axiomatic model-theoretic homotopy theory, stable homotopy theory, chromatic homotopy theory. Recently, we got a topological version of algebraic geometry, namely spectral algebraic geometry which is proved useful in studying topological modular forms.

But one may wonder what is it for? Those are really fancy and sometimes beautiful tools, but what are exactly the questions modern algebraic topology seeks to answer? Because It feels it's really not part of topology anymore, it's more as topology now is a small part of algebraic topology/homotopy theory.

So, I would like to hear about goals and perspectives of modern homotopy theory from those working on it. I hope this question might be useful to someone else.

Answer 1

While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still an especially important place in our heart. Especially when we say algebraic topology instead of homotopy theory. This says that while all of category theory and all of homological algebra belongs to the study of $(\infty, 1)$-categories, this is not where our aim is.

The roots of our subject lie in the study of nice spaces like manifolds. Important questions are:

1. Can we classify manifolds up to some equivalence relation?
2. Can we understand maps between manifolds?

The coarsest useful equivalence relation for the classification of manifolds is bordism and this is also the basis of most other classification results (those using surgery theory). Computing bordism groups was an important topic in earlier algebraic topology and was done succesfully for some flavors rather early ($\Omega_O$, $\Omega_U$, $\Omega_{SO}$,...). But one of the most important variants, both theoretically and from the viewpoint of clasification of manifolds, is framed bordism. By an old theorem by Pontryagin, the framed bordism groups are isomorphic to the stable homotopy groups of spheres, connecting it to the second question.

One can say that much of algebraic topology was invented or can be used to study the stable homotopy groups of spheres. One of the most recent spectacular advances in algebraic topology was the solution of (most of) the Kervaire invariant 1 problem by Hill, Hopkins and Ravenel about framed manifolds/stable homotopy groups of spheres. They used a tremendous amount of stuff to solve this classical problem: equivariant topology, chromatic homotopy theory, spectral sequences, orthogonal spectra, abstract homotopy theory, ...

Likewise topological modular forms $tmf$ have important applications to the stable homotopy groups of spheres and also to string bordism. And to really understand $tmf$, you have to study some spectral algebraic geometry.

I do not want to say that all of algebraic topology still directly aims at classical questions. As soon as we see an interesting structure, we also study it for its own sake; new phenomena need explanations and developing abstract frameworks is also fun. But like in the relationship between mathematics and physics, sharpening our tools and exploring by pure curiosity can be quite useful for the classical questions. When people replaced older, in some aspects more clumsy models of spectra by symmetric and orthogonal spectra, they probably didn't have in mind any direct applications to framed manifolds. But what Hill, Hopkins and Ravenel did would have been much harder without these tools in their hands.

Answer 2

Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant replacements, etc) in many different settings. In some of these settings (e.g. homological algebra), it's not a huge surprise that you can do so. But settings include

• Representation theory (via the stable module category)
• Algebraic geometry (via motivic homotopy theory)
• Graph theory (via work of Bissen and Tsemo)
• Category theory (via work of Rezk, among others)
• Universal algebra (via colored operads and PROPs)
• Mathematical physics (via TQFTs)
• Dynamical systems (via Gaucher's work on flows)
• Computer science (via work of David Spivak, among others)

All these settings form model categories, so you can carry out your favorite constructions from homotopy theory. In each setting, the methods of abstract homotopy theory have been used to prove new theorems. I think as a field we need to market ourselves a bit better, and show the people in these areas that we're not trying to force them to use our language, but we are offering them some tools that they might find useful.

When one works in the abstract setting, it's common to try to specialize to get results in some or all of these areas. Most abstract homotopy theorists do still care about topological spaces and spectra, so will include examples specializing to those settings in their papers. In terms of the questions abstract homotopy theory asks, it seems they come in several flavors:

1. Finding new examples to add to the list above, or proving examples encode the same homotopy theory (if they are model categories, this is asking for a Quillen equivalence).
2. Taking problems in areas on that list and trying to solve them using the abstract methods. For example: the cobordism hypothesis, Milnor's conjecture, Vandiver's conjecture, Happel's book on representation theory.
3. Computing homotopy groups, algebraic K-theory groups, Tate cohomology, etc. Here, the abstract setting is what allows you to bring over tools of spectral sequences and other transfinite computational processes.
4. Developing additional unifying structure, e.g. finding the right way to encode some definition or theorem, so that it recovers seemingly different things on items in the list above, proving that they are in fact special cases of the same general concept.

In terms of the difference between model categories and $\infty$-categories, it seems the field has remained fairly civil about the fact that there are two settings in which you can "do abstract homotopy theory." Each has strengths and weaknesses. I use model categories because it feels closer to my intuition from spaces, because all the examples I care about are model categories, because I have more tools to use in that setting (e.g. fibrant/cofibrant replacement), and because I've always been able to make things work in that setting. If I was ever trying to prove something and got sidelined by technical details I might switch to $\infty$-categories if that would make life easier.

Answer 3

In response to Ryan Budney's comment, let me try to say something about topological data analysis, and other recent applications of algebraic topology outside of traditional mathematics.

Applied Algebraic Topology has been around in various forms for many years. I first learned about it in my training in computer science from Rob Ghrist's work. In fact, I wrote an MO answer back in 2011 about his work. The point seems to be efficiently computing sheaf cohomology, with applications in electrical engineering. Why sheaves? I'll illustrate with an example. All over the country a bunch of moving cell phones are trying to connect to a bunch of cell towers. The regions those towers can reach form a cover of your space. If a cell phone is in a place not covered by any tower, it's bad news, and you want to be able to detect that. Homology helps, since it finds holes. More importantly, if a cell phone is in an intersection, then it has many towers to talk to, and that can cause interference. Sheaf cohomology comes into play here, and can help you design better systems, detect interference issues, and even create coding schemes to fix the confusion interference can cause.

More recently, Gunnar Carlsson's group at Stanford (and his company) has been using algebraic topology to compute on data (my interest is that I mostly teach statistics nowadays). It's called Topological Data Analysis. If you've ever taken a basic statistics course, you know we often use linear regression, i.e. find the best fitting line and use it to make predictions for x values where we don't have any data. If the data is not linear, we transform it (via logs, square root, etc) to make it linear. But that's just because linear things were easy back in the days before computers. Nowadays you could use computational software to run much more complicated regressions. It's just as easy now to fit a curve (e.g. polynomial regression) as a line, since both involve pushing a button on any statistical software. Why stop at curves? If your data comes in the shape of a manifold, why not try to fit a manifold to the data, and use that manifold to predict values of the dependent variable for various combinations of values of the independent variables. Topological data analysis strives to give you the tools to do this. On a more basic level, persistent homology lets you detect holes in your data, by which I don't mean missing values, but rather actual regions where data is not coming to you because it's not being generated there.

As a silly example, think of taking a picture of Lake Geneva at night. You'd probably see lots of lights ringing the lake, but none inside it. The data here are the lights, and the fact that there are no lights coming from the lake is telling you that something is not there. Similarly, you could imagine taking a picture of the sky and noticing dark spots as a way to find satellites. The examples Gunnar's group has produced are much more useful and less contrived. I believe several have to do with breast cancer data. If you google, you'll find lots of slides of talks he's given, replete with examples.

Persistent homology works by considering all possible covers of your dataset by balls of radius r drawn around the data points, as r varies. It's best to imagine 2 dimensional data where you roughly see the shape of a circle. When r is very small, the cover is entirely disconnected. When r is very large, you're probably looking at a bunch of intersecting balls, with way too many overlaps to tell you much. But for some value of r in the middle, you get a connected shape that looks roughly like $S^1$. The balls form a simplicial complex, and that's how the computations are done. When the balls form many disconnected components, $H_0$ has large dimension. Once they coalesce into a connected component, $H_0$ is $\mathbb{Z}$ and (in the circle example) $H_1$ is also $\mathbb{Z}$. It remains $\mathbb{Z}$ as $r$ gets larger and larger, till r becomes so large that the union of the covering balls forms a disc rather than a circle (up to homotopy). The word "till" in the last paragraph is why it's called "persistent" homology. One way to visualize how the homology groups change with r is to write them as barcodes, where the left-to-right axis is r and the number of bars is the dimension. When you see a long barcode, that's telling you a feature of your data that is persistent even as r varies, e.g. a hole.

There are also applications of topological data analysis (TDA) to Machine Learning, Clustering, and Classification. A simple example is barycentric clustering, which is something like a souped-up, topological version of k-means clustering. Gunnar's group has more complicated examples that have been useful in identifying previously unknown associations, that were later backed up with theory. A common problem is dividing a dataset into distinct pieces, e.g. via Support Vector Machines. Basically: if your dataset can be separated by a hyperplane then you do so. If not, you transform to a higher dimensional space where it can be and then separate it there (equivalently, you find a separating sheet or surface). I am hopeful that the methods of TDA can be used to provide improved separation algorithms.

More recently, Kathryn Hess has gotten involved with applications of algebraic topology to neuroscience. This is related to both work of Ghrist and work of Carlsson, but different from both. Now the game is to discover how information travels across the network of neurons in your brain. Working with rats, you can stimulate the brain and empirically measure how electricity moves. You can then try to uncover traits of the network based on which pathways are being used frequently, and you can try to figure out what determines the path taken and what difference the path taken makes. I know less about the work Hess is doing here, but I know it has to do with computing Betti numbers and using them as invariants. Carlsson also has work related to neural networks (I seem to recall hearing that rats have a Klein bottle in their brain, but have no idea why), but I think it has a different flavor.

In a similar vein, there was an AMS Special Session at the 2012 Joint Math Meetings entitled Generalized Cohomology Theories in Engineering Practice. I only got to one talk in that session (about K-theory as an invariant of some engineering system), but perhaps googling the speakers would lead to more useful applications.

Incidentally, there are also algebraic topologists working in graph theory, to use algebraic topology to make new graph algorithms. Certainly computing $H_1$ is a way of detecting cycles. From what I understand, the algorithms produced so far don't do much that is new and interesting, and are much less efficient than existing algorithms. There are also people studying random simplicial complexes in the way that random graphs have been well studied. For an example, see this paper on arxiv and follow the references. Finally, there are people writing down effective algorithms to compute in simplicial sets, e.g. here. All of this may bear fruit, as we learn better how to model the world using simplicial complexes and simplicial sets, and as we find ways to wrangle data into forms where our tools can be used to attack it.

Answer 4

I'm going to give an algebraist's perspective. First let's discuss homological algebra (which has roots in topology). There's a quote (attributed, I think, to Connes) that a great mystery of homological algebra is the power afforded by the formula $d^2=0.$ This formula is indeed mysterious and the resulting study of chain complexes yields powerful results. But an even more powerful formula, that people rarely think about in this way, is $$a+d = b+c \implies (a,b)\sim (c,d).$$ This fundamentally powerful equivalence relation on pairs of natural (or real positive) numbers yields wonderfully versatile number systems, which have much better properties than the original semigroups. But mathematicians don't think about $\mathbb{Z}$ or $\mathbb{R}$ as equivalence relations on pairs: rather, we think of them as fundamental objects in their own right: the equivalence relation is simply a way to access them. The same can be said about differential graded categories: the whole formalism of projective resolutions and quasiisomorphisms is just a way to access objects like the Serre dualizing sheaf that are fundamental, and that the more rigid formalism of abelian categories is insufficient for. In this sense, the category of complexes is just one particularly elegant choice for "softening" the abelian category (this is a term I first heard from Kontsevich): others exist as well. (See e.g. You Qi's hopfological algebra.)

The modern interest in topology, as I see it, comes from the fact that it turns out that some fundamental objects interesting to all mathematicians, including algebraists (e.g. K-theory spectra, tangent complexes in characteristic p, certain cyclic and motivic constructions) are not accessible through pure algebra, but can be "evaluated" on topological spaces, and can then be reconstructed from these evaluations using techniques related to modern homotopy theory. It is somewhat miraculous that so many constructions interesting to algebraists have good formulations in terms of something topologists had thought about for a long time, but this is just one of those miracles we have come to expect from any elegant mathematical theory.

The important questions of homotopy theory from this point of view are of the form "what are the right definitions, when do they work, and how do we compute with them". For example, suppose $A$ is some algebraic gadget (algebra, Lie algebra, etc.) Consider the group of automorphisms of structure, $Aut(A)$. In really nice cases, this is a Lie group, it's locally determined by its Lie algebra (of derivations), and it is the stabilizer of a point in a well-behaved classifying orbifold. It turns out that if one allows to supplement geometric ideas by topological ones, the class of "really nice cases" expands dramatically. But one needs to understand the appropriate meaning of the words "Lie group", "local", "Lie algebra", etc., and what one can do with them. This is done in very strong generality in Jacob Lurie's work on Moduli spaces.

Answer 5

Very many mathematical notions (groups, rings, algebras, etc) form categories.

Similarly, a number of mathematical notions (spaces, spectra, derived schemes, etc.) form $(\infty,1)$-categories. Their study is what modern algebraic topology is about.

To summarize:
Modern algebraic topology is the study of everything that forms an $(\infty,1)$-category.